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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> iff <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> )</span><br/>

<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> implies <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> implies <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> or <font color="Olive" title="b1">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">y</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<a NAME="E2:7_1"/>
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font> or  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:7_1"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:7_1"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="ref" href="yellow_7.html#T1" target="_self" title="YELLOW_7:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>  holds <br/>( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> or <font color="Olive" title="b1">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">y</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D29" title="WAYBEL_0:def.29">WAYBEL_0:def 29</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<a NAME="E3:7"/>
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  or <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  )
 
<b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:7"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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